Functions $f$ that satisfy $f(x) + f(1/x) = \rm constant$ for each $x$.

Question

Hello I am thinking about all the functions that satisfy $$ f(x)+f(1/x) = C $$ for each $x$. The constant $C$ is the same for all $x$ in the Domain.

It is clear that $\log_a(x)$ works for all possible bases. Can you think of another function?

Answer 1

For any function $h(x)$, let $f(x)=A[h(x)-h(1/x)]+B$. Then, $$ f(1/x)=A[h(1/x)-h(x)]+B\implies f(x)+f(1/x)=2B. $$